0^^i''^}^!^ 


AN  EXTENSION  OF  THE 
STURM-LIOUVILLE  EXPANSION 


// 


/ 


By 
CHESTER  CLAREMONT  CAMP 


Respectfully  submitted  to  the  Faculty  of  the  Graduate  School 

of  Cornell  University  in  partial  satisfaction  of  the 

requirements    for   the  degree  of 

Doctor  of  Philosophy 


AN  EXTENSION  Ol^  THE 
STURM-LIOUVILLE  EXPANSION 


By  ^  '' 

CHESTER  CLAREMONT  CAMP 


Respectfully  submitted  to  the  Faculty  of  the  Graduate  School 

of  Cornell  University  in  partial  satisfaction  of  the 

requirements    for  the  degree  of 

Doctor  of  Philosophy 


ACXCHANQ^ 


Reprinted  from  The  American  Journal  of  Ma-bhkm.^tics,  ypJ.XLIX,  No.  3,  Januarj',  1922 

AN   EXTENSION   OF   THE   STURM-LIOUVILLE   EXPANSION. 

By  Chester  Cl.\remont  Camp. 

Introduction. 
In  1836-7  C.  Sturm's  formal  development  of  a  more  or  less  arbitrary 
function  j{x)  in  terms  of  solutions  of  the  self-adjoint  equation 


d  /     du\ 
TxVTx) 


du 


and  the  Sturmian  boundary  conditions 


+ 
+ 


9^0 
9^0 


au{a)  +  a'u'(a)  =  0, 
fiu{b)  +  /3'i/'(6)  =  0, 

was  considered  by  J.  Liouville,*  who  undertook  the  problem  of  showing 
that  the  series  converges  and  that  its  value  is  fix).  His  work  is  important 
although  it  did  not  satisfy  all  the  requirements  of  modern  mathematical 
rigor,  but  in  two  remarkable  papers  A.  Kneserf  completely  settled  all  the 
more  fundamental  questions  concerning  the  development.  It  remained 
for  HaarJ  several  years  later  to  give  the  solution  finality. 

In  1908  Birkhoff  |1  extended  the  theory  not  only  to  equations  of  the 
7ith  order  of  the  form 

but  to  systems  no  longer  self-adjoint  and  conditions  no  longer  Sturmian. 

The  theory  is  capable  of  extension  in  several  directions.  B6cher§ 
considered  a  system  of  two  homogeneous  linear  differential  equations  of  the 
first  order  and  studied  the  roots  of  a  solution  without  regard  to  boundary 
conditions.  Schlesinger^  took  a  system  of  n  linear  equations  of  the  first 
order  with  coefficients  which  are  rational  in  x  and  obtained  the  asymptotic 
forms  for  a  solution. 

The  object  of  this  paper  is  to  discuss  an  extension  of  a  problem  recently 
considered  by  Professor  Hurwitz,  namely  the  simultaneous  expansion  of  two 

*  Liomnlle's  Journal,  Vol.  1  (1836),  p.  253;   Vol.  2  (1837>,  p.  16  and  p.  418. 

t  Malh.  Ann.,  Vol.  58,  p.  81  and  Vol.  60,  p.  402. 

t  Goettingen  dissertation  (1909)  reprinted  in  Malh.  Ann.,  Vol.  09  (1910),  p.  331.  Also 
a  second  paper,  Mcdh.  Ann.,  Vol.  71  (1911),  p.  38.  See  also  Mercer,  Phil.  Trans.,  Vol.  211 
(1911),  p.  111. 

II  Trans.  Amer.  Malh.  Soc.,  Vol.  9,  p.  373. 

§  Trans.  Amer.  Math.  Soc.,  Vol.  3  (1902),  p.  196. 

%  Math.  Ann.,  Vol.  63  (1907),  p.  277. 


SOTToG 


26  .Cave*  o4?i  Extmsiprupf%e  Sturm-Liouville  Expansion. 

functions /(a;),  g{x)  in  terms  of  solutions  of  the  system 

du       J-  -. 

^  =  [X  +  a{x)2v, 

^  =  -  [X  +  bix)']u, 

and  the  boundary  conditions 

au(a)  +  fiv{a)  =  0, 
a'u(b)  +  ^'v{b)  =  0. 

Although  this  system  is  simple,  it  is  not  self-adjoint  and  the  functions 
a{x),  b{x)  are  any  two  real  functions  which  possess  continuous  second 
derivatives. 

The  extension  I  have  made  is  that  resulting  from  the  substitution  of  the 
more  general  linearly  independent  boundary  conditions 

aiu{a)  +  i3i«(a)  +  yiu(b)  +  ^iv(b)  =  0, 
a2u(a)  +  ^2v(a)  +  72^(6)  +  82v(b)  =  0, 

in  which  the  coeflficients  are  such  as  to  satisfy 


Oil    /3i 
(X2     1S2 


7i     5i 

72       ^2 


7^0. 


I  wish  to  acknowledge  my  indebtedness  to  Professor  Hurwitz  for  his 
constant  interest  and  careful  direction  as  well  as  for  his  recent  article  as  a 
basis  for  research. 

For  further  references  the  reader  is  pointed  to  the  following  by  Bocher: 
Encyklopsedie  der  Math.  Wiss.,  Band  II,  Heft  4,  pp.  437-463;  Proceedings 
of  the  Fifth  International  Congress  of  Mathematicians,  Vol.  I  (1912), 
pp.  163-195. 

Section  I.     Preliminary  Lemmas. 


Consider  the  system 


Ni(uv\)  =  u'(x)  -  [X  +  a{x)2v(x)  =  0, 
N2{uv\)  ^  v\x)  +  [X  +  b(x)2u{x)  =  0, 


(1) 


with  the  boundary  conditions 

(2) 
in  which  X  is  real  or  complex;   a{x),  b{x)  are  real  functions  possessing  con- 


Ui(uv)  =  aiuiO)  +  PMO)  +  7MI)  +  5iKl)  =  0, 
U2{uv)  =  a2u{0)  +  /32KO)  +  72W(1)  +  82V{1)  =  0, 


Camp:  An  Extension  of  the  Sturm'LioupiUe.Expansiim.'. 


27 


ai     /3i     7i     5i 

0C2     02     72     ^2 

is  of  rank  two  and 

(ocfi)  =  (76)  9^  0 

where 

/ 

"2    P2 ; 

tinuous  second  derivatives  for  0  ^  x  ^  1 ;    and  the  coefficients  of  u,  v,  in 
(2)  are  real  constants  such  that  the  matrix* 


(3) 


The  sohition  u{x)  =  t(x)  =  0  is  called  the  trivial  solution  of  (1),  (2). 
All  other  solutions  are  termed  non-trivial. 

Studying  non-trivial  solutions  of  (1),  (2)  we  derive  the  following  lemmas. 
Lemma  I :  //  (um,  Vm)  and  {iin,  Vn)  are  two  solutions  of 


then 


Ni(ukVk\k)  =  NiiukVkXk)  =  0, 


[,'Um(x)Vn(x)  —  Vn{x)Vm{x)'y^=o 

=  (X„ 
By  hypothesis 


K)  J'o'[Um{s)Un(s)  +  »m(«)»n(*)]^J?.       (4) 


Wm(a^)  —  D^  +  a(x)']i}m(x)  =  0, 

i?m(a*)  +  [Xm  +  b{x)2um{x)  =  0, 

u'„(x)  —  [X„  +  a{x)2  Vn{x)  =  0, 

'  v:(x)  +  lK  +  b(x)'}u^ix)  =  0. 

If  we  multiply  these  equations  by  Vn{x),  —  «„(.r),  —  Vm{x),  and  Um(x) 
respectively,  add,  and  integrate  from  0  to  x,  we  obtain  the  equation  (4) 
required. 

Lemma  II:  //  (u,  v)  satisfies  Ni  =  <p,  N2  =  yp,  where  (p,  \J/  are  functions 
of  X  continuous  0  ^  a:  ^  1,  and  (U,  V)  satisfies  Ni  =  N2  =  0  for  the  same 
value  of  X,  then 


\:u{x)V(x)  -  Uix)v(x)yo  =  Jl'LV(s)<p(s) 
By  h}T)othesis 


U{s)rl^{s)']ds. 


(5) 


u'(x)  -  [X  +  a(x)-]v(x)  =  <p{x), 
v'{x)  +  [X  +  h{x)'}u{x)  =  yl^{x), 
V'{x)  -  [X  +  a{x)']V{x)  =  0, 
V'{x)  +  [X  +  h{x)']U{x)  =  0. 


•  If  the  matrix  is  of  rank  less  than  two  the  conditions  (2)  are  not  independent.  The 
condition  (o^)  =  (78)  is  somewhat  analogous  to  the  condition  of  self-adjointness  for  single 
equations  of  order  two.  The  case  in  which  {(x0)  =  (76)  =  0  can  obviously  be  reduced  to 
the  problem  considered  by  Professor  W.  A.  HiuTvitz. 


28  .Caiwp:  ^n  Extension  of  the  Sturm-Liouville  Expansion. 

Multiplying  respectively  by  V(x),  —  U(x),  —  v{x),  u(x);  adding  and 
integrating  as  before  we  get  equation  (5)  above. 

Lemma  III:  Two  pairs  of  functions  u{x),  ti{x)',  U(x),  V(x);  satisfying 
Ui  =  1/2  =  0,  satisfy  also  the  relation 


[_u{x)V(x)  -  v(x)U(x)J  =  0 
when  (3)  is  satisfied. 
We  are  given 

«iw(0)  4-  ^iHO)  +  Tiw(l)  +  511^(1)  =  0, 

a2u{0)  +  /32KO)  +  72W(1)  +  521^(1)  =  0, 

aiU{0)  +  ^iF(O)  +  yrU{l)  +  5iF(l)  =  0, 

a2U{0)  +  /32F(0)  +  72U{1)  +  62F(1)  =  0. 

Since  (aj8)  9^  0, 

1     l7iM(l)  +  Wl),     /3i 


(6) 


w(0)  =  - 


and 


v{0)  =  - 


(afi)  \  T2W(1)  +  ^2^1);    ^2 
1     !  ai,     Tiw(l)  +  8iv{l) 


{al3)\a2,     72^(1)4-52^1) 


also  17(0),  F(0)  are  similar  functions  of  ^7(1),  F(l). 

Substituting  these  values  in  (6)  we  get  for  the  condition  necessary  that 

(a/5)2  =  (5/3)  (7a)  +  (7^)(a5)  =  {a^){y8) 

which  is  evidently  true  by  (3). 

Lemma  IV:    //  (ui,  vi),  (u2,  V2)  are  two  linearly  independent  pairs  of 
functions  of  x,  continuous  0  =  x  =  1,  then 


G2^ 


(11)     (12) 
(21)     (22) 


is  real  and  greater  than  zero,  where 

(rs)  ^  yoTwr(«)ws(*)  +  Vr{s)vs(s)']ds 


and  u{x) 

represents  the  conjugate  of 

u{x). 

Obviously  G2  = 

-G2, 

i.e.,  G2  is  real. 

Let 

p{x)  = 

(11)     ur{x) 
(21)     U2{x) 

J 

q{x)  ^ 

(11)     ^i{x) 
(21)     t2{x) 

} 

i.e., 

p{x)  =  CiWi(.r)  +  C2W2(a;), 

q{x)  =  ci 

^A^)  +,C2»2(i 

.). 

Camp:  An  Extension  of  the  S^t7mrLiouviUe.E::tpp4i^on/  29 

Calculate 

^,r   /  N-  /  X    .      /  N     /  x-ij         1(11)     (1^)1        {0,i{k=l, 
fo^Lp(x)u.(^)  +  ry(.)..(.)]rf.  -  I  ^2i;     l^k)  r  1  (?„  if  ^  =  2. 

Then 

fo'i\pi^)\'+  \q(x)\')dx^  fo'\j>i^)p(^)  +  Q(^Vjix)y^ 

=  Cifo'\JP(x)Mx)  +  q{x)vi(x)ya-  +  royo'Qj(.r)M2(.T)  +  q{z)v2{x)yx 
=  C2G2  ^  0. 

If  (?2  =  0  and  C2  =  P2  7^  0,  then  p(ar)  =  g(a;)  =  0  and  (wi,  rO,  (1*2,  ^'2) 
will  be  linearly  dependent. 

If  6*2  =  0  and  Cz  =  0,  then  since 

C2  ^  (11)  ^  yoHki(a:)|^+  |^i(a:)|2)rfa:  =  0, 

Wi  =  n  =  0  and  they  will  again  be  linearly  dependent,  thus  violating  the 
hypothesis. 
Hence 

(?2  >  0  (7) 

since  Co  >  0,  and  the  lemma  is  proved.* 
Lemma  V:  //  (3)  is  satisfied,  then 

[(76)  +  (a/3) J  ^  l(ay)  +  W8)J  +  [(a5)  +  (7^) J.  (8) 

(8)  can  easily  be  shown  to  be  equivalent  to 

,    (78)2  +  (afi?  ^  ("7)'  +  W^y  +  (ccdy  +  (7/3)^  (9) 

But  • 

(a7)2+(W^2(a7)(/98), 
and 

Hence  the  right  member  of  (9)  is 

^  2(ay)(fi8)  -  {a8)(^y) 
^  2(a^)(76),  or  by  (3) 
^  {a^y  +  (75)^ 

,     Since  (9)  is  true,  (8)  is  also. 

'  Lemma  VI:  //  (3)  holds,  it  is  impossible  for  the  coefficients  in  (2)  to  satisfy 

{ay)  +  m)  =  0, 
(a5)+(7^)  =  0.  ^ 

•The  lemma  admits  of  the  following  generalization:    If  (ui,  Vi),  (uj,  Vt),  •••  («„,  f«) 
are  n  linearly  independent  pairs  of  functions  of  x,  continuous  0  ^  2  ^  1,  then 

M  (11)     (12)     •••     (In) 
^    _  :  (21)     (22)     • .  •     (2n) 

I  (nl)     (n2)     • • •     (nn)  \ 
is  real  and  greater  than  zero,  n  being  a  positive  integer. 


30  ^AM^*  <^^  EgdeV'Siqn  of  the  Sturm-Liouville  Expansion. 

By  (3) 

(a^)  -  (75)  =  0. 

If  then  we  assume  (10)  true,  by  squaring  and  adding  all  these  equations, 
we  have 

(ayy  +  m'  +  (a8y  +  (7/3)^  +  (a^Y  +  {y8y  +  2(^7)  (^S) 

+  2(a8)(7/3)  -  2(«^)(75)  =  0. 

The  sum  of  the  last  three  terms  of  the  first  member  vanishes  identically 
so  that  if  (10)  holds,  each  of  the  six  determinants  must  vanish.  Since 
this  violates  (3)  the  proof  is  complete. 

Section  II.     Properties  of  Solutions  of  the  Homogeneous  and  Non-homo- 
geneous Systems. 
Lemma  I:    A  necessary  and  sufficient  condition  for  the  existence  of  a 
solution  {u,  v)  of  (1),  (2)  is  that  the  determinant 


D{\)^ 


UiiUiVi)        Ui{u-iV2) 
UiiUiVi)        U2{llllV2) 


(11) 


vanish  for  some  value  of  X,  where  (ui,  vi),  {ui,  V2)  are  solutions  of  (1)  defined  by 

um  =  1,     vM  =  0, 

W2(0)    =    0,  i)2(0)    =1.  ^       ^ 

By  the  existence  theorem  we  know  that  a  solution  either  of  (1)  or  of  the 
corresponding  non-homogeneous  system 

Ni  =   <p,  N2  =  rp, 

where  <p,  x}/  are  functions  of  x,  continuous  0  ^  a;  ^  1,  will  be  entire  in  X, 
provided  it  is  defined  by 

'w(O)  =  ci, 
v(0)  =  Co, 

in  which  the  c's  are  independent  of  X.     Clearly  D{\)  will  also  be  entire  in  X. 
Any  solution  of  (1)  is  expressible  in  the  form 

u(x)  =  u(0)ui(x)  +  v(0)u^{x),  ,.„. 

v{x)  =  u{0)vi(x)  +  v{0)v2{x). 

In  order  that  u(x),  v(x)  satisfy  (2)  it  is  necessary  and  sufiicient  that  u(0), 
v{0)  be  determined  so  as  to  satisfy 

Ui{u)  =    Ui(UiV,MO)  +    Ui(U2V2)v(0)  =  0,  ,j^. 

U2{U)  =    U2(UiVi)u(0)  +    U2{U2V2)V(0)   =   0, 

or  if  u{x),  v(x)  is  to  be  a  non-trivial  solution,  the  determinant  of  coefiicients 
of  w(0),  v(0),  which  is  P(X),  must  vanish.     This  is  obviously  also  sufficient. 


Camp:  An  Extension  of  the  Sturm-Liouville  Expajision.^  31 

Lemma  II:  The  roots  of  D{\)  are  all  real  and  the  solutions  of  (1),  (2) 
may  he  taken  as  real. 

IjQt  X  =  Xjt  be  a  root  of  D{\).  Then  (uk,Vk)  satisfies  Ni(ukVk\k)  =_0, 
NiiukV^y^k)  —  0  and  {uk,  Vk)  will  satisfy  the  same  conditions  for  X  =  Xjt. 
Hence  by  Lemma  I,  Section  I 

{uk'Ck  —  UkV)P^»  =  (Kk  —  \k)f'Ouk{s)  |2  +  \vk(s)  \^)ds. 

Moreover  {uk,  Vk)  satisfies  Ui(u)  =  U-i{u)  =  0  so  that  by  Lemma  III, 
Section  I  _ 

(Xt-x,)y;Hk*|2+  \vk\'')dx  =  {). 

If  Xjt  7»^  Xfc,  {uk,  tk)  is  a  trivial  solution  which  we  have  excluded.  Therefore, 
Xjfc  =  Xfc  and  Xjt  is  real. 

Since  each  equation  of  the  system  (1),  (2)  is  linear  and  a{x),  b{x)  are 
real,  any  solution  of  the  system  if  complex  may  be  broken  up  into  real  and 
imaginary  parts  which  will  separately  satisfy  the  same  system.  When 
D{\)  vanishes  the  ratio  of  u{0)  to  i'(0)  or  of  v{0)  to  m(0)  is  determinable*  and 
by  the  existence  theorem  the  solution  u(x),  v{x)  is  then  uniquely  determined 
except  for  a  constant  factor.  Hence  the  coefficients  of  the  real  and  imag- 
inary parts  constitute  two  solutions  which  are  linearly  dependent.  It  is 
therefore  clear  that  we  may  restrict  ourselves  to  real  solutions  of  the  system. 

A  similar  discussion  will  show  that  we  need  consider  only  real  solutions 

of  the  system 

Ni  =  <p,        N2  =  ^,  (15) 

where  (p,  \f/  are  real  functions  of  x  and  X  is  restricted  to  real  values,  a  restric- 
tion which  we  shall  henceforth  make. 

A  value  of  X  which  makes  Z)(X)  vanish  is  called  a  principal  parameter 
valve  and  the  corresponding  solution  of  the  homogeneous  system  (1),  (2) 
is  known  as  a  principal  solution. 

Since  D{\)  is  an  entire  function  it  cannot  have  more  than  a  finite 
number  of  roots  in  any  finite  interval  of  the  X-axis,  so  that  we  may  designate 
the  roots  of  D{\)  by  \„,  n  =  0,  ±  I,  ±  2,  ■■■. 

Theorem  I :  A  sufficient  condition  for  the  existence  of  a  solution  (m,  v)  of 
(15),  (2)  analytic  for  all  real  finite  values  of\is  that 

'1""m     t<1k  =  0  (16) 


X 


for  every  (w„,  Vn)  satisfying  (1),  (2)  /or  X  =  X„,  n  =  0,  zb  1,  ±  2,  •  •  •. 
We  shall  consider  in  turn  three  kinds  of  values  of  X: 
Case  I:  Those  for  which  the  matrix  of  (11)  is  of  rank  two; 

*  If  the  matrix  of  (11)  is  of  rank  zero  (Case  III  below),  then  this  is  found  by  the  eval- 
uation of  an  indeterminate  form. 


32  Camp  I  An  Extension,  of- the  Sturm-Liouville  Expansion. 

Case  II :  Those  for  which  the  matrix  is  of  rank  one  (at  X  =  X„) ; 
Case  III:  Those  for  which  it  is  of  rank  zero  (at  X  =  X^). 
Take  a  solution  of  (1), 

U(x)   =   CiUi(x)  +  C2U2(x), 
V{x)    =   CiViix)  +  C2V2(x), 

which  is  identical  with  (13),  provided  w(0)  =  ci,  v(0)  =  c^.  This  solution 
will  satisfy  (2)  if  (14)  is  satisfied  and  will  be  non-trivial  provided  Ci,  c^  are 
not  both  zero.  For  Case  I  at  least  two  of  the  elements  of  Z)(X)  must  be 
different  from  zero.  Without  loss  of  generality  assume  that  f/i(?^2«2)  9^  0, 
then  by  continuity  it  will  not  vanish  for  values  of  X  nearby.  Put 
Ci  =  —  Ui(u2V2),  C2  =  Ui{uivi).  Then  Uiiuv)  =  0,  and  U2iuv)  =  D{\). 
Define  another  solution  (C/,  F)  of  (1)  by 


U(x)  =    —   C2Ui(x)  +  CiU2ix), 
V{x)  =   —   C2Vi(x)  -h  CiV2(x), 

Then  (u,  v),  {U,  V)  will  be  linearly  independent  since 


(18) 


W  = 


liiO)      <0)    _         c,     C2  2   ,     2  ^  n 

—  =  C]  +  Cj  >  0. 

—   C2       Ci 


U(0)     F(0) 

Consider  a  solution  {u,  v)  of  (15)  defined  by 

u(x)  =  wo(a-)  +  biu{x)  +  b2U(x), 
v(x)  =  vo{x)  +  biv(x)  +  62^(0-), 

where  (uq,  vq)  is  a  particular  solution  of  (15)  such  that 

Uo{0)  =  i'o(O)  =  0. 
Then 

1  U2{uv)   =  U2(.uoVo)  +  b,U2(vv)  +  bolMUV), 
and  (u,  v)  will  satisfy  (2)  if 


(19) 


(20) 


62  = 


&i= 


W      ' 

Ul(UoVo)U2iUV)   -    U2(UoVo)W 
WU2(UV) 


(21) 


,    ,     ,     U,(UoVo)U{x)  Ui(UoVo)U2(UV)  +    U2(UoVo)W      ^    ^ 

u{x)  =  u^ix)  H ^ ^^^^^ u{x), 

^    ^     ,      Ui(UoVo)V{x)  L\{UoVo)U2iUV)  +    U2iUoVo)W  ^^^^ 

v(x)  =  Vo{x)  -\ ^ ^^^^^^ v{x), 

since  U2{uv)  =  D(k). 

Hence  if  the  matrix  of  (11)  is  of  rank  two,  (22)  determines  a  solution 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 


33 


^,  v)  analytic  in  X  for  that  range  for  which  Ui{u2V2)  9^  0.  For  other  ranges 
of  vklues  of  X  within  which  U\{u-yV2)  vanishes  we  choose  in  its  place  another 
element  of  the  determinant  P(X)  which  differs  from  zero  throughout  that 
range.  This  process  determines  h\,  hi  in  a  different  fashion.  Clearly  since 
the  roots  of  UiiihVi)  and  of  every  other  element  of  D{\)  are  isolated,  each 
element  being  entire  in  X,  the  solution  {u,  v)  determined  in  either  way"  for 
points  of  the  X-axis  at  which  UiiihVi)  9^  0  and  some  other  element,  e.g., 
UiiuiVi)  7^  0,  will  be  identical  since  there  is  one  and  only  one  such  solution 
by  the  existence  theorem.     .*.  The  proof  is  complete  for  Case  I. 

Case  II:  Let  us  first  show  that  no  root  of  Z)(X)  is  double  for  this  case. 
Assuming  that  Uiiv^ih)  5»^  0  for  X  =  Xn  and  defining  Ci,  Cj  as  before,  we  have 
by  differentiating  as  to  X,  since  u{x),  v{x)  of  (17)  are  entire  in  X  and 

satisfy  (1): 

[Niiv^vO^)  =  vix), 

[Niiu^vM^  -u{x).  ^^^^ 

If  Z)(X)  has  a  double  root  X  =  X„,  then  for  this  value  of  X 

Uiinv)  =  0,     C/2(wx«x)  =  0;        Ni(uv\)  =  NiiuvX)  =  0; 
Uiiuv)  =  0,     Uiiu^v^)  =  0. 

Whence  by  Lemmas  II,  III,  Section  I 

fo%'u{s)J-\-[v{s)J)ds^O. 

This  is  impossible  since  w(0)  =  —  UiiUiVi)  ^  0.  Therefore  Z)(X)  has  no 
double  root.; 

For  values  sufficiently  near  Xn,  Z)(X)  7^  0  and  the  solution  (w,  i)  of  (15), 
(2)  analytic  in  X  is  given  by  (22).     And  by  Lemma  II,  Section  I 

luo(x)v(x)  -  u(x)vo(x)']^  =  fo'[.v(s)<p(s)  -  uis)xf^{s)ys,  (24) 

liK^ixWix)  -  U(xMx)J,  =  fo'lV(s)<p(s)  -  U{s)^P{s)-]ds.         (25) 

The  left  members  vanish  at  the  lower  limit  since  Uo(0)  and  Do(0)  are  equal 
to  zero,  hence  solving  we  obtain 

u(x) 
u{s) 
vis) 

v(x) 
u(s) 
vis) 

X„,  w(ar)  approaches  the  value 


iMiix) 


voix)  = 


'4: 


Uix) 
Uis) 
Vis) 

Vix) 
Uis) 
Vis) 


0     I 

<pis)  \ds, 

^(*)l 

0 

<pis)    ds. 

rPis) 


(26) 


(27) 


If  X 


.^(,)+?^C,(,)--^) 


^LUiilM>Vo)U2iUV)  +   U2iUoVo)W2 


2>'(X) 


(28) 


*=*, 


34  Camp:  An  Extension  of  the  Sturm^Liouville  Expansion. 

provided  only 

UiiuoVoW^iUV)  +  U2(uoVo)W:\,^,^  =  0  •  (29) 

since  Z)'(X)  9^  0. 

Again  if  (29)  holds,  vix)  will  approach  a  limit  as  X  -►  X„.  In  such  a 
case  (u,  v)  will  be  continuous  if  put  equal  to  the  limit  approached  at  X  =  Xn 
and  therefore  analytic  for  all  X  in  the  interval. 

Since  W=  -  UiiUV),  from  (26),  (27)  it  follows  that  (29)  is  equivalent  to 


£ 


0 


TiC^(l)+5iF(l),     UriUV) 
72t/(l)+52F(l),     U^iUV) 
u{s)  U(s)  <p(s) 

<S)  Vis)  ^(*):x=Xn 


7iw(l)+5i2)(l),     UiiUV) 

T2W(1)  +  52<1),        U2iUV) 


ds=0. 

n 

(30) 


The  first  element  of  the  large  determinant  is  the  same  as 

I  7iw(l)  +  5iKl),    yiUil)  +  5iF(l) 

I  72W(1)  +  52^1),     72^^(1)  +  52^(1) 

aiw(O)  +  fiMO),     aiUiO)  +  /3iF(0) 
«2W(0)  +  i82K0),     a^UiO)  +  i82F(0) 

since  for  X  =  X„,  Ui(uv)  =  Uziuv)  =  0,  or  simplifying  further,  this  element 
reduces  to  (y8)W  -  (afi)W  =  0. 

Thus  a  sufiicient  condition  for  the  existence  of  a  solution  {u,  v)  of  (15), 
(2)  analytic  for  all  X  when  D{X)  is  of  rank  one  is  that  (16)  be  satisfied  for  all 
n  such  that  D{\n)  =  0,  and  the  theorem  is  proved  for  Case  II. 

Case  III.  When  every  element  of  Z)(X)  vanishes  for  some  X  =  \k,  it 
is  obvious  that  D'(Kk)  also  vanishes.  Let  us  show  that  Z)"(Xjk)  7^  0  for 
such  a  case. 

By  the  same  reasoning  as  was  used  to  prove  D'(X)  ?^  0  for  X  =  Xn  in 
Case  II  we  show  that  if  C7i(wix«ix)  and  U2{uixV\^  both  vanish  at  X  =  Xt, 
then 

f^\[u,{x)J  +  lv,{x)J)dx  =  0, 

which  is  impossible  since  wi(0)  =  1.  Hence  ?7i(wix«ix)  and  U2(uixVi\)  can- 
not both  vanish  at  X  =  X^.  Similarly  ?7i(w2x^2x)  and  U2(u2\V2\)  cannot  both 
vanish  there.     Without  loss  of  generality  we  may  assume 

C7i(M2x«'2x)  ^0,  X  =  X,.  (31) 

Now  define  a  solution  of  (1)  by 

u{x)  =  Ui{x)  -h  —ihi^), 

""'  (32) 

v(x)  =    Vi(x)  +  -  Viix), 
C2 

and  choose 

£l_  _  UljUiVi)  _ 
C2  Ui{2l2V2) 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion.  35 

For  values  of  X  sufficiently  near  Xjt,  Ui(ihVo)  7^  0,  otherwise  (31)  would 
be  violated  by  Rolle's  Theorem.     Hence  for  all  X  in  the  interval  considered 

Consequently  for  all  such  X,  UiiuxV^)  =  0. 
Again 

At  X  =  \k, 

And 

For  X  =  \k, 

D"{\)      _^  D{\)  C7l(W2XXl'2Xx)  +  D'(\)  U,{u,y) 


U2(UxTx)   =    - 


Ui{'lh\V2x)  2Ui{UiV2)Ui{v^ix'02\) 


This  vanishes  if  Z>"(X)  =  0  at  X  =  Xjt.  Applying  the  argument  a  third 
time  we  obtain 

/oKw'  +  ''^)dx  =  0, 

which  is  absurd  since  by  (32)  w(0)  =  1 . 

Since  for  Case  III  when  X  =  X^-  every  solution  of  (1)  satisfies  (2),  if  we 
define  a  solution  of  (15)  by 

fu(x)  =  Vo(x)  +  hiiti(x)  +  biU^ix),  .^„v 

1  V{X)  =    Vo(x)  +  biViix)  +  b2V2(x),  '     ^•^'^^ 

where  Wo(0)  =  ro(0)  =  0  defines  a  particular  solution  of  (15),  then 
U\{ut)  =  Ui{uoVq)  and  U2{uv)  =  U2(uoVo).  If  these  vanish  fei,  62  will  be 
arbitrary.  Again  by  using  results  similar  to  (24),  (25),  (26),  (27)  we  have, 
since 

wi(0)     ri(0)  i  _  , 

«2(0)       tJ2(0)|  ^' 

/•I   7iWi(l)  +  5i»i(l),     7iw^(l)  +  5x^2(1),     0      I 
UiiltcVo)  =  —    I  ui(s),  v^{8),     (p(s)    ds, 


U2(.lliiVo) 


«/0 


72Wl(l)  +  52»l(l),       72W2(1)  +  52^^(1),        0 

«i(*),  Ms),    <p{s) 

^i(*),  «2(*),    i{s) 


ds. 


For  X  =  Xfc,  (wi,  ti),  (m2,  t>2)  are  solutions  of  (1),  (2).     Hence  if  we  assume 


36  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

that  (16)  holds  for  all  solutions  (Un,  Vn)  of  (1),  (2),  then  (w,  v)  will  satisfy 
(2)  for  bi,  62  arbitrary. 

We  wish  to  choose  values  for  them  such  as  to  make  the  solution  (w,  v), 
now  analytic  X  5^  Xjt,  continuous  at  X  =  Xa;. 

Since  every  solution  of  (1)  may  be  expressed  as  a  linear  combination  of 
ui,  Ui,  vi,  V2,  we  may  put  (33)  in  the  form 

fu(x)  =  Uq(x)  +  diu(x)  +  dzMx),  ,oA\ 

in  which  {u,  v)  is  defined  by  (32)  and  ei/c2  has  the  same  value  as  before. 
{u,  v)  and  (^2,  ^2)  are  linearly  independent  since 


w(0)      v{0) 

W2(0)        «2(0) 


1     ,Ci/c2 

0       1 


5^0. 


As  TJ\{uv)  ^  0,  we  have 

Ui{uv)  =  Ui{uoVo)  +  d^tfiiu^v^), 
which  will  vanish  for  all  X  in  the  interval  considered  if 

Again 

Uziuv)  =  UiiuoVo)  +  diUiiiiv)  +  diUiiuiiOi)  =  0, 
provided 

7     ^    _   U2(UqVo)Ui(U2V2)  +    Ul(UoVo)U2(U2V2)   ^ 
Ui(U2V2)U2{uv) 

For  X  ?^  X;fc, 

Ui{U2V2) 

and 

J     _    U2(UoVQ)Ui('WiV2)  -h    Ui{UoVo)U2(lhV2) 

* W) 

Hence  for  X  =  Xa;  in  a  sufficiently  small  interval,  Z)(X)  5^  0  and  Ui(u2,  V2)  9^  0, 
«(x)  =  «„W+  ^^("»'^) V^{MV.(v4oi) V.{v.^i> ^(^^ _  Mf4^(^).     (35) 

-L'(X)  Ui{ll2V2) 

If  we  let  X  approach  X^  we  get  as  the  value  approached  by  the  right  member: 

rr  r  \  :SSr2  C  ^2  (uqVo)  Ui  {U2V2)  +  Ui  (tioVo)  U2  {U2V2) ] 

u,{x)  -  ^^i^  U2{x)  +  ^^(.r)  ^-^ , 

Ui{U2xV2x)  D"(K) 

X  =  Xit. 
Evidently  u{x)  approaches  a  limit  since 

£1  _^  _  UijihxVix)  _ 

C2  UiiU2\V2x) 


Camp:  An  Extension  of  the  Stnrm-Liouville  Expansion.  37 

Also 

—  [,U2('UqTo)Ui(U2V2)  +    Ui(UoVq)U2{'U2V2)']  =0,  X  =   Xfc, 

ah. 

since  each  factor  of  both  terms  vanishes  there.     Also  Z)'(Xt)  =  0. 

Hence  u{x)  approaches  a  limit  as  X  -*  Xjt,  and  similarly  v{x)  does  also. 
Thus  we  have  a  solution  (u,  i)  analytic  for  all  X  in  a  small  enough  interval 
about  X  =  Xjk,  which  satisfies  (2).  By  extending  the  reasoning  as  in  Case 
I  the  proof  is  completed. 

Theorem  II:  If  for  n  =  0,  ±  1,  ±  2,  -  -  ■ 


f 


Unis)       <p(s) 

i^n(s)     rp(s) 


ds  =  0, 


then  (p(x)  =  \p(x)  =  0. 

The  proof  will  be  merely  outlined  here  since  it  is  given  in  full  in  a  recent 
article  by  Professor  W.  A.  Hurwitz. 

Let  the  solution  of  (15),  (2)  shown  by  the  previous  theorem  to  exist  be 
represented  by 

u(x)  =  yo{x)  +  \yi(x)  +  \^y2(x)  +  •  •  •, 
.  v{x)  =  zoix)  +  \z,{x)  +  \'Z2{X)  +  . .  • .  ^'^^^ 

Since  X  is  restricted  to  real  values  we  have  but  real  functions  of  x  with 
which  to  deal.  Putting  (36)  in  (15),  (2)  and  equating  coefficients  of  X* 
we  get  sets  of  differential  equations  and  boundary  conditions  satisfied  by 
{ym,  Zm),  m  ^  0,  1,  2,  •••.  By  eliminating  a(x),  b{x)  from  two  different 
sets  and  using  Lemma  III,  Section  I,  one  gets 

/oiymyn  +  Z,nZn)dx  =    (JVy,n-\yn+l  +  Z„^lZn+l)dx  =    W^^ny 

where 

^r  =    Jo^iyryo  -f  ZrZo)dx. 


Then  from  the  inequality 

2 


y„^i{x)    ym^i(^) 

Vmr-lix)       ym^liO 


+ 


+ 


y«_l(x)       2„_i(^)  I 

2Wfi(a:)    ym+iiO 

Zm^l{x)      yn^iiO 


+ 


Z„^l{x)       2W+-l(^) 


^0,     (37) 


for  x  and  ^  in  the  interval  from  zero  to  one,  by  integrating  as  to  x  and  ^ 
successively  from  0  to  1  and  simplifying,  we  have 

W2n^2W2r^2  "    Wl^  ^  0.  (38) 

Next  comes  the  lemma  that  some  W^2ik  =  0.     Assume  no  ^'2*  =  0.     Multi- 
plying the  series  of  (30)  by  y^ix),  zq{x)  respectively,  adding  and  integrating 


38  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

we  obtain  the  uniformly  convergent  series 

This  converges  absolutely  as  also  does 
Using  (38),  i.e. 


(39) 
(40) 


forX 


=v 


Wo 


W, 


W2^Wi  =  We' 

in  this  we  get  an  absurdity 


^0 1  +  \Wo\  H"  I  PFo I  +  •  •  •  convergent.     Thus  the  lemma  is  proved, 


I.e. 


foKyl  +  4)dx  =  0,         or         yk{x)  =  Zk{x)  ^  0.  (41) 

Thence  from  the  differential  equations 

yk-i(x)  =  yk-2{x)  =  ...  =  yQ{x)  =  0, 
and  similarly  for  z  and  also  4/(x)  =  (p{x)  =  0.     Q.e.d. 

Section  III.    Asymptotic  Formulae. 
Theorem  III:    For   |X|  large  a  solution  of  (1)  defined  by  u{0)  =  a> 
v{0)  =  j8  takes  the  form 


u{x)  =  a  cos  ^  +  /3  sin  ^  +  0  I  r- 1' 
vix)  =  jS  cos  ^  —  a  sin  ^  +  0  I  ^  1' 


(42) 


and  its  partial  derivatives,  the  form 


where 


^x(^)  =        l^x  cos  ^  —  ax  sin  ^  +  0 


Vx{x)  =  —  ax  cos  ^  —  jSa;  sin  ^  +  0 


^  =  X.T  +  hfo'Lais)  +  b(s)2ds. 


& 


(43) 


(44) 


I  give  an  outline  of  a  proof  analogous  to  that  used  in  Professor  Hurwitz's 
recent  article. 
Assume 


u(x)  =  U+(l+  ^\  {a  cos  ^  +  /3  sin  0, 
v(x)  =  F  +n  +  -j^)()3  cos  I  -  a  sin  ^). 


(45) 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 
Putting  in  (1)  we  get 


39 


(46) 


where  (p,  \f/,  <px,  ^x  are  0(1)  as  regards  X. 

Using  as  multipliers  cos  "Kx,  —  sin  \x,  adding,  integrating  from  0  to  x> 
and  combining  this  with  the  result  of  repeating  the  procedure  with  sin  Xx, 
cos  \x  as  multipliers  we  get 


(47) 


where  F,  G  and  the  K's  are  0(1),  and  F^,  Ox,  K^  are  also.  , 

■••     UM  =  f  +  Jl'lKnUis)  +  K,2V{s)  +  KzzUM  +  K^,VMlds, 


(48) 


in  which  H,  J,  and  the  K*s  of  (48)  are  0(1).     By  the  process  of  successive 
approximations  we  obtain  from  (47),  (48), 

Uix)  =  0  (^^^  V(x)  =  0  0'  UM  =  0  (~j'  VM  =  0  0-      (49) 

From  (45),  (49)  we  see  that  (42)  is  true.     Also  by  differentiating  (45)  as 
to  X  and  using  (49)  the  rest  of  the  proof  is  obvious. 

Corollary:  For  |X|  large  Z)(X),  D'{\)  take  the  forms 


Z)(X)  =  ^1A'  +  B'  sin  [^(1)  +  ^']  +  C -{- 0  (^\ 
D'{\)  =  <A^  +  52  cos  K(l)  +  ^]  +  0  (i  V 


where 


A  ^  {ay)  +  (i95),         B  =  {ab)  +  (7/3),         C  =  2(a^), 
and  ^p  is  defined  by 


cos  (f  = 


sm  <p 


B 


(50) 


(51) 


(52) 


By  Lemma  VI,  Section  I,  A  and  J5  cannot  both  vanish.     From  Theorem  III 


40  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

ui{x)  =  cos  ^  +  0  (  -•  J  , 

vi(x)  =  —  sm^  +  0(-j> 

thix)  =  sin  ^  +  0  f  -  j  » 

V2(x)  =  cos  ^  +  0  (  -  J  • 

Putting  these  in  (11)  we  get 

D(X)  =  A  sin  ^(1)  +  B  cos  ^(1)  +  M)  +  {y8)  +  o(^\ 


(53) 


(54) 


Again  using  the  asymptotic  values  for  uix{x),  vix{x),  etc.,  from  (53)  in  the 
identity 


we  derive 


+ 


C^2(Wix?Jlx)        U2{'lh\V2)d 


D'(X)  =  A  cos  ^(1)  -  B  sin  ^(1)  +  0  (^\ 


and  (50)  follows  by  (3),  (51),  (52). 

To  show  D{X)  has  roots,  no  matter  how  large  |X|  is,  consider  the 
Theorem  IV:  If  In  is  a  principal  parameter  value  for  the  system  (1)  and 

the  boundary  conditions 

jaMO)  +  iSiKO)  =  0, 
|Tiw(1)+  8M1)  =  0, 


(55) 


then  there  exist  exactly  two  roots  of  D(\)  in  the  intervals: 

Case  I.     {hp,  hp+i),  p  =  0,  db  1,  ±  2,  •  •  • ,  if  P21  >  0  for  X  =  /i; 
Case  II.     (/2JH-1,  hp+i),  p  =  0,  ±  1,  =b  2,  •  •  •,  if  P21  <  0  for  X  =  h* 
Define 

'Bi(x)  =        aiu(x)  +  I3iv{x), 

Bzix)  =       a2u{x)  +  ^2'v{x),  (56) 

P\{x)  =  —  yxu{x)  —  8iv(x),  ; 

Piix)  =  —  y2u(x)  —  hvix), 


and  determine  two  solutions,  [ui{x\),  vi{xKy\,   [u2{xk),  V2{x\)'},  of    (1) 
such  that 

fBn(O)  =  0,         52i(0)  =  1, 

Ui2(0)  =  1,         ^22(0)  =  0; 


(57) 


in  which  the  second  subscript  refers  to  the  solution  involved.     Without 

*  I  have  followed  the  method  of  proof  used  by  Professor  Birkhoff  in  an  article  published 
in  the  Transactions  in  1909  entitled,  "Existence  and  Oscillation  Theorem  for  a  Certain 
Boundary  Value  Problem." 


Camp:  An  Extension  of  the  Stiirm-Liouville  Expansion.  41 

loss  of  generality  we  may  assume  that  the  coeflScients  ai,  /3i,  71,  5i  have 
been  divided  by  a  constant  such  that 

(aiS)  =  (76)  =  1.  (58) 

Then  we  prove  the 

I^EMMA  I:   At  a  simple  value  X„  of  X,  Z)(X)  changes  sign  in  such  a  way 
that  D'{\)  has  the  sign  of  —  Pn  or  P22,  where 

jPn  =  -  TiWi^l)  -  5i?Ji(l),  ff.Q. 

IP22  ^    -   72W2(1)   -   52^2(1).  ^^^^ 


The  solutions  (ui,vi),  (ui,V2)  defined  by  (57)  are  linearly  independent  since 

=  (afi)  =  -  1  (60) 


wi<0)     tJi(O) !  ^    -  ^1  ai 

W2(0)       ^2(0)  I  ^2       -   Ci2 


and  hence  by  a  relation  similar  to  Abel's 

\u{x),    v{x)  I  =  constant  7^  0. 
Thus 

Bn(x)B22(x)  -  Bi2(x)B2i(x)  =  -  1,        0  ^  a;  ^  1.  (61) 

Likewise 

Pn(x)P22{x)  -  Pnix)P2iix)  =  -  1,         0  ^  a:  ^  1.  (62) 

Also  any  solution  of  (1)  may  be  written 

u{x)  =  CiWi(ar)  +  C2i^(a-), 

V{X)   =    CiViix)  +   C2V2{X). 

If  this  is' to  satisfy  (2)  we  must  have 

£>(X)  =  L  "  p'^'  ^"  p'H  =  0.  (63) 

i  i   —   i^21>  "~   -t  22  I 

We  have  shown  that  the  necessary  condition  for  a  double  value  of  X  is 
that  Z)(X)  be  of  rank  zero,  i.e., 


Pn  =  P22  =  0, 
P21  =  Pn  —  1. 


(64) 


This  is  obviously  also  sufficient. 
Since 

Ni(uivi\)  =  Niiuivik)  =  0  (65) 

and 

^■l(Wlx^lxX)  =  vi,        NiiuiyVixX)  =  -  u,  (66) 

by  Lemma  II,  Section  I, 

wix^i  —  wifix  =  yo'(w?  +  T}'i)d^-  (67) 

Similarly 

«2x^  —  ihV2x  =  fo'(nl  +  vDds.  (68) 


42  Camp:  An  Extension  of  the  Sturm- Liouville  Expansion. 

Again  from 

NiiuiViX)  =  NiiuiViX)  =  0  (69) 

and  (66)  by  the  same  Lemma 

Uix'Ci  —  U2V1X  =  So'iuiUi  +  V1V2)  ds.  (70) 

Similarly 

U2kVi  —  U1V2K  =   J'o''iUiU2  +  ViV2)ds.  (71) 

From  (67),  (70) 

wu(a:)  =  fo'^liul  +  vl)u2(x)  —  (uiU2  +  ViV2)uiix)2ds.  (72) 

Similarly 

^ik(x)  =  JTLi'^l  +  ^1)^2(3:)  —  (W1W2  +  viV2)vi{x)']ds.  (73) 

Also  from  (68),  (71) 

^\(a^)  =  JVL(uiU2  +  viV2)u2{x)  —  (u]  +  vl)ui{x)']ds,  (74) 

V(^)  =  yo'C(wiW2  +  «i»2)i'2(a:)  —  (w2  +  vl)vi{x)2ds.  (75) 

Now  from  (62),  (63) 

D(K)  ^  Pu  +  P21  -  2.  (76) 

Henceby(72),  (73),  (74),  (75),  (76) 

D'(K)  =  Pi2x  +  P2U 


or 


D'W   =   fo'lP22u\  +    (P12  -   P2l)WlW2  -   Pliwi 

+  P22'vl  4-  (P12  —  P2i)^'i^2  —  Pnvr\ds. 


(77) 


The  integrand  consists  of  two  quadratic  forms  whose  discriminant  is 

(P12  -  P21)'  +  4PnP22  (78) 

or  by  (62) 

(P12  +  P2i)^  -  4.  .  (79) 

This  vanishes  when  D(X)  does.  For  a  simple  root  of  D  (X)Pii,  P22  cannot 
both  vanish  since  then  by  (78)  and  (63),  (64)  would  be  satisfied.  Again 
neither  form  can  vanish  because  {ui,vi)  and  (^2,^2)  are  linearly  independent. 
Thus  the  Lemma  is  proved. 

Lemma  II:  At  a  double  valu£  of  X,  say  Xa,  D{\)  maintains  a  negative  sign. 

From  (64) 

APii  =  Pii,       AP22  =  P22,       1  +  AP12  =  P12,       1  +  AP21  =  P21,     (80) 

in  which 

P=  P(\-\-  AX),         X  =  X,.  (81) 

Then  from  (62) 

P11P22  —  P12P21  =  ~  1 
or 

AP11AP22  -  (1  +  APi2)(l  +  AP21)  =  -  1,  (82) 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion.  43 

and  by  (76) 

AD  =  APi2  +  AP21  =  APuAP-n  -  AP12AP21.  (83) 

Clearly 

APu  =  -  yiAuiil)  -  5iAri(l), 

and  so  for  the  others.     Also 

Ni(Au,  Av,  X)  =  A\v, 

NiiAu,  Av,  X)  =  —  AXw. 

Awi    Aiiz    Aci    Avi 
Using  these  for  (?^i,»i),   Cm2,»2)  we  get  results  for     .^,  -rr-,  -rr-,  -rr 

which  differ  little  from  the  right  members  of  equations  (72)  to  (75).  Putting 
these  values  in  (83)  and  omitting  in  each  term  infinitesimals  of  order 
greater  than  two,  we  obtain 

A^  =  \lfo'(uiU2  +  ViV2)A\dsJ  -  f,Ku\  -f  v\)A\dsf,\u\  +  v-^AUs. 

By  Lemma  IV,  Section  I,  AD  <  0,  since  the  solutions  are  real,  and 
D{\)  preserves  a  negative  sign  at  X  =  X^. 

Lemma  III:  Z)(X)  has  the  same  sign  as  P21  at  the  values  \  =  In,  n  =  0, 
±  1,  rfc  2,  •  •  •,  unless  P21  =  1,  when  D{\)  =  0. 

Since  Wi(0)  =  —  /3,  ci(0)  =  a  (Cf.  (60)),  (ui,Vi)  is  the  only  solution 
except  for  a  constant  factor  which  satisfies  (55),  (1)  for  X  =  Iq,  l^i,  Z±2, 
•  •  •  but  for  no  others.  (See  Professor  Hurwitz's  article.)  These  values 
separate  the  X-axis  into  the  intervals 

•  •  •  '(l-n,  U+l),        •  •  •    (/-I,  k),       (/o,  /l),        •  •  •    (In,  In+l),        '  '  '  (84) 

By  (62)  at  X  =  one  of  these  values,  say  Ir, 

P12P21  =  L  (85) 

And  by  (76) 

D{lr)  =  ^  (1  -  ^2l)^         r  =  0,  ±  1,  ±  2,  . . ..  (86) 

And  the  rest  of  the  proof  is  evident. 

Professor  Hurwitz  has  shown  that  Pn  has  no  double  roots,  also  that  for 
the  system  (1)  and 

:?:S::: 

P21  has  no  double  roots.  Hence  Pn,  P21  change  sign  when  they  vanish. 
Again  since 

P11P2IA  ~   PlU^2l  =   Wl»lA  ~  ^iWiA^x-l. 

by  (67)  this  cannot  vanish  but  is  negative.  From  these  facts  it  is  easy  to 
show  that  the  roots  of  Pn,  P21  separate  each  other  as  well  as  do  those  of 


44  Camp:  An  Extension  of  the  SturTn-IAouville  Expansion. 

Pn\,  -Pau-     Clearly  then  P21  alternates  in  sign  at  the  values  •  •  •  I-2,  Z_i,  lo, 
l\,  h,  •  • '  and  we  have  two  cases : 

Case    I:  |  ^^^^  <  q  at  X  =  /o,     /.2,  •  •  •  }  ^'^'^  ^"  >  0  for  X  =  /:; 

Case  II:   \  7^., .  -^  n    x  \        ?*  '  7^  '  f  when  P21  <  0  for  X  =  h. 

\  lf{\)  =  U  at  A  =  /o,     1^2,  •  •  ■  J 

There  must  obviously  exist  roots  of  Z)(X)  as  follows: 

In  Case  I — at  least  two  values  X„,  X„+i  in  each  double  interval 

ihp,  I2P+2),  p  =  0,  d=  1,  =t  2,  •  •  •, 

such  that 

hp  <C  X„  =  hp+i  =  Xti+i  <C  hp+2  (88) 

and  at  least  one  such  that 

h^\<l2. 

In  Case  II — at  least  two  values  X^,  X^+i  in  each  double  interval 

(hp-i,  hp+i),  p  =  0,  ±  1,  ±  2,  •  •  •, 

such  that 

hp—l  K  \n  =  hp  "^  \n+l  <  hp+l-  (89) 

To  show  that  there  are  exactly  two  roots  in  (kp,  I2P+2)  we  make  use  of  the 
separative  property  of  the  roots  of  Pn,  P21  and  their  derivatives.  Obviously 
there  must  be  an  even  number  of  roots,  since  we  count  a  double  root  as  two. 
If  a  double  root  occurs  it  must  fall  at  kp+i  bj^  (64)  and  Lemma  III.  Then 
by  Lemmas  II  and  I  there  can  be  no  root  elsewhere.  If  there  is  no  double 
root,  there  must  be  less  than  four  roots,  for  otherwise  there  would  be  at 
least  two  in  one  of  the  intervals  (hp,  hp+i),  (hp+i,  hp+2),  which  violates 
Lemma  I. 

Similarly  we  can  show  that  there  is  exactly  one  root  in  the  interval 
(/i,  h)  and  exactly  two  for  Case  II  in  the  intervals  (hp-i,  hp+i)  provided  we 
count  a  double  root  at  h  once  each  in  the  intervals  (h,  h),  (h,  h)-  Thus  the 
Theorem  is  proved. 

Corollary.    For  |X|  large,  0(1/X„)  =  0{\lmr),  \n\  large. 

Professor  Hurwitz  has  shown  that  for  |X|  large.  In  =  nir  -\-  di  -\-  0{ljn), 
where  di  is  a  constant.  Thus  the  intervals  (/„,  4+2)  for  | X | ,  \n\  large  are 
of  length  27r  to  within  0{\Jn).  In  each  of  these  intervals  for  n  odd  or  even 
according  to  Case  I  or  II  by  the  Theorem  there  exist  just  two  values  X^, 
Xm+i  of  X  which  are  roots  of  D(\).  Since  in  any  finite  interval  of  the  X-axis 
there  are  but  a  finite  number  of  roots  Ir  or  X,„,  for  |  n  \  large  enough  we  have 

X  =  ln+8  —  k,  0  ^    \k\    ^  T, 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion.  45 

where  *  is  a  finite  integer  positive  or  negative,  provided  we  begin  to  count 
the  /'s  and  X's  so  that  /o  =  0,  Xo  =  0  and  /_i  <  0,  X_i  <  0.     Clearly  then 

X„  =  nir  +  *x  -  ^  +  01  +  a  [^\  (90) 

and  since  mtt  is  the  dominant  part, 


Ky^K^J'"-^"- 


Theorem  V:  //  {un,  Vn)  corresponds  to  Xn  defined  as  in  the  premoiis  Coroll- 
ary, then  this  solution  of  (1),  (2)  takes  the  asymptotic  form 


iin{x)  =  sin  \jiTX  -\-  P{x)']  +  0  (  -  |, 

Vnix)   =   COS  {jlTTX  +  P(x)]  +  ^  (  "    ). 


(91) 


when  \n\  is  large. 

By  the  Corollary  of  Theorem  III  D(K)  will  have  a  root  when 


or  when 

-  C 


(92) 


^(1)  +  ^  =  sin-»-p=^+  O^iJ,  (93) 


since  by  Lemma  V,  Section  I,  \C\  ^  '^A}  +  B^,  and  by  Theorem  IV  the 
right  member  of  (92)  must  be  numerically  ^  1,  also. 


and  for  X  =  X„, 

»a)-"(i)-»(-;)- 

Hence  from  (93),  since  by  (58)  C  >  0,  Z)(X)  =  0  when 
and  for  vi  odd 

Xn+i  =  (m  +  l)7r  -  i/'  -  fn^  — dx—  (p-\-  0[-\, 

in  which  ^  is  the  angle  in  radian  measure  between  0  and  7r/2  inclusive 


46  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

whose  sine  is  |  C  |  /  -ylAF-^-B^.     Clearly  (94)  may  be  expressed  as 

\n=mT-\-(-  1)-+V-  fo'"^  dx-  ^+0(y\,^l\n,     (95) 
where  \n\,  \Xn.\  are  large  and  m  =  n  -{-  s.     From  (42) 
u(x)  =  sm(^-eo)  +  o(^Y 

'0{x)  =  cos  {^  -  do)  +  0  (^Y 

provided  we  choose 

a  =  —  sin  ^0,         ^  =  cos  ^o- 


This  is  always  possible  if  we  divide  w(0),  v{0)  by  ^[^(0)]]^  +  C2'(0)]]^  and 
call  the  quotients  a,  ^  respectively.     Then  by  (95) 

Unix)  =  sin  \jmrx  +  62X  +  ^fo^'Z^^is)  +  his)^^  —  Bo']  +  0  I  -j , 

Vn{x)  =  cos  [m,irx  +  d2X  +  ^JV[_a{s)  +  b{s)2ds  —  5o]  +  ^  (  -  J, 
where 

62  ^  (-  1)"*+^  -  fo'  ^  dx-  <p 

and  the  formulae  (91)  follow,  provided  we  define 

P(x)  ^  d2X  +  STX  +  fo  "^^^  "^  ^^'^  ds  -  do, 

since  m  =  ti  +  *,  ^2  =  —  ^-  +  0i  as  shown  bj^  (90) . 

Corollary:  If  we  normalize  (un,  Vn),  it  will  have  the  same  form  (91). 

Section  IV.     Expansion  Theorems. 
Theorem  VI:    If  two  otherwise  arbitrary  functions  f(x),  g(x)  satisfy  (2) 
and  have  continuous  second  derivatives,  0  ^  .r  ^  1,  then  they  are  expansible 
in  the  form 

+  00 
fix)  =   ^CnUnix), 

":r  (96) 

gix)   =    ^CnVnix), 
n=  — 00 

where 

Cn  =   y*oT/(^)Wn(^)  +  gix)Vnix)~\dx,  (97) 


and  {Un,  Vn)  represents  the  set  of  normal  orthogonal  solutions  of  (1),  (2). 

In  case  \k  is  a  double  value  we  count  it  as  two  and  take  any  two  linearly 
independent  normal  orthogonal  solutions  for  that  value  of  X. 


Camp:  An  Extension  of  the  Siurm-LiouviUe  Expansion.  47 

Let  us  first  show  that  two  solutions  {um,  Vm),  {Un,  Vn)  of  (1),  (2)  for 
diflferent  values  of  X  are  orthogonal,  i.e., 

f,'[Um{s)u^{s)  +  Vm{s)Vn{s)yS  =   0.  (98) 

It  is  only  necessary  to  apply  Lemmas  I,  III,  Section  I,  and  divide  by 
X^  —  X„.     Obviously  they  are  easily  normalized  by  dividing  in  the  case  of 

(Mm,  O  by  <flJul^^vl)dx. 

If  we  assume  the  series  of  (96)  to  be  uniformly  convergent,  then 

yoT/(^)*^(-^)  +  g{x)Vm{x)yix  =   Cm 

and  the  series  become 


S   ^<n(a')yo^[/(^)«7,(ar)  +  g{x)vn{x)'yix, 

+  00 

S      1^n{x)Jl^[J'(x)Un(x)  +  g(x)Vn(x)Jdx. 


(99) 


To  study  these  we  assume  |X„|  >  |a(a')  |,  also  >  J6(a;)|.     We  have 

y'nix)   —    (X„  4-   a)Vn{x)   =    0, 
V'nix)   +    (Xn  +    b)Ur,{x)   =    0. 


Then 


f(x)      . 
So^j{x)Un{x)dx  =  —  Jo^  \   4-  b  ^'•(^)^^. 

r    MvMT  ,   ^,  ,  ,d[_j(x)_i 

.   =L"XTtJ„  +  ^«^^"(^)5^[x;:tk^)J^^ 

r    fvn  T ,  ^.  K(x)  dv  f  -] 

L      X„+6jo"^*^°  K  +  adxlK  +  br''' 

But  since  by  the  Corollary  of  Theorem  IV  0  (  —  j  =  0  (  —  )  =  0  (-\ 

for  \n\  large,  we  have 

d  (      fix)      \         fix)         f(x)b'(x) 
dx\K  +  b{x))  ^  K+b      iK  +  bJ  ^  ^  /J_\ 
\n  +  a{x)  \n+a  \n- J 

Similarly 
Hence 

'"^  ^  [  x;h=^  ~  rf^  Jo  +  ^  (^0 


48  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

or 

c„  =  ^  Igun  -  fVnJ  +0(^\^'  (100) 

Since  by  hypothesis  /,  g  satisfy 

UiU,  g)  =  u,u,  g)  =  0,  (loi) 

by  Lemma  III,  Section  I,  Cn  =  0{l/n^)  and  each  series  in  (99)  converges 
uniformly.  If  f{x),  g{x)  are  expansible,  we  get  the  series  (99)  for  them. 
To  show  that  these  series  converge  uniformly  to  f{x),  g{x)  respectively, 
we  define 

+  00 

F{x)    =   Z)  Un{x)  fQ^[^f{x)Un{x)  +  g{x)Vn{x)yix, 

n——QO 

Gix)  =   Zl  'Vn(x)fQ^[_f(x)Unix)  +  g(x)Vn{x)2dx. 

n=  — 00 

Then  formally  we  have 

Jo^[_F{x)Unix).+  G(x)Vn{x)'}dx  =   Jh^[_f{x)Un{x)  +  g{x)Vn(x)']dx 

or 

JhHZFix)  -f{x)2un{x)  +  iGix)  -  g{x)-]vn{x)]dx  =  0, 

w  =  0,  ±  1,±2,  .-.. 
If  now  we  define 

F(x)  -fix)  ^  rPix),    .     Gix)  -  gix)  =  -  ^(x), 

then  by  Theorem  II  we  have 

Fix)  ^  fix),        Gix)  ^  gix),  q.e.d. 

Theorem  VII:    If  fix),  gix)  do  not  satisfy  (2)  hut  possess  continuous 
second  derivatives  as  before,  the  series 

fcoMo(a:)  +  [ciUiix)  +  c_iW_i(.t)]  +  [c^u^ix)  -f  c_2W_2(a:)]  +  •  •  •,     qq2) 
\cQVQix)  +  [ciViix)  +  c_i«_i(a:)]  +  [cic^ix)  +  c-^v-^ix)'}  +  •  •  •, 

converge  uniformly  to  fix),  gix)  forO<€^a:^l  —  e<l,  where  e  is  an 
arbitrarily  small  positive  number  and  iun,  Vn),  Cn  are  defined  as  before. 
We  have  from  (100) 

CnUnix)   =    -y—  \igix)Unix)   -  fix)Vnix)Jii  -\-  0  \—A  ' 


Hence  by  (91)  for  ln|  large 


,    .            (-   l)''KiUnix)   -  KoUnix)     , 
CnUnix)   =    — h 


"(i) 


where 

Xi^5r(l)sinP(l)-/(l)cosP(l), 
Ko  =  giO)  sin  P(0)  -  /(O)  cos  P(0). 


But 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion.  49 

Again 

,,       (-  l^Kiu^jx)  -  Kou^jx)   .    ^/  1  \ 

c^u.M  = zr^ +  ^  U  j  • 

Hence 

Cn^niir)  +  C-nU-n(x)    =        ^ ^  J  •-""  ~   """-J" 

?/„(.r)  —  u-„(x)  =  sin  [nxa-  +  P(a-)]  —  sin  [—  wttx  +  P(a-)]  +  ^M  ~  ) 
=  2  sin  (nirx)  cos  P(a-)  +  ^^  (  "  )  ' 

•'•      r„u„(x)  +  C^U-„{x) 

T^,  s  r  rr  /      ,s    sin  nxa:       _,  sinnxj;~|    ,    ,.  /  1  \ 
=  2  cos  PW  [  A,(- D- -^^  -  A.  ^^  J  +  0  (;^)  • 

^,           •        v>  (~  1)*  sin  nTTj;        -^  sin  nirx  ..       .       , 

Ihe    series    2^ '      Z^ converge    uniformly,   for 

,;  =  l  nw  ^x         71T 

0<€^x^l  —  €<  1,  and  since  we  get  a  similar  expression  for 

the  rest  of  the  proof  is  like  in  the  Theorem  above. 

Section  V.     Further  Results  on  the  Distribution  of  Principal  Parameter 

Values. 
Witliout  the  use  of  Theorem  IV  we  can  show  that  for  the  case  in  which 


\C\  <  ^IA^+  B\ 

D{\)  has  exactly  two  roots  in  each  interval 

Q2A-  —  l)7r,  (2k  +  l)7r],   \k\   large  and  integral, 

which  have  the  asymptotic  form  (95)  for  \n\,  |X|  sufficiently  large.     With- 
out loss  of  generality  assume  (58),  then  by  definition  C  =  2,    Then 

^1A^+~B'  -2  =  E>0.  (103) 

By  (50) 

D{\)  =  Vl2+^sin  [^(1)  +  ^]  4-  C  +  ^' '         0^\K\  <  M, 

for  1 X 1  large,  or 

D{\)            ...          C        _,K' 
=  sin  A  +  •  ,       +  -T- ' 


VlH^'  ^A'  +  B'       ^ 

where 

A  =  ^(1)  +  <p.  (104) 

For  A  =  (2A-  —  l)7r  or  2A-ir,  and  [Xj  >  Xj/, 


50  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

For  A  =  (4:k  -  l)7r/2,  and  |X|  >  X^, 

^(^)  =-i+-^i^+g-'=  -^  +g:<o. 


Let  Xi  be  the  larger  of  \m,  X^,  then  for  |X|  <  Li,  D(\)  must  have  at 
least  two  roots  in  the  intervals  (21-  —  l)7r  <  X  <  2kT.  Since  for  2kw  ^  A 
^  (2^  +  l)7r,  sin  A  ^  0,  D{\)  has  no  root  there.  To  show  that  there  are 
exactly  two  roots  in  the  previous  interval  we  use  the  fact  that  Z)(X)  =  0 
only  when 

sin  A  =  —     ,  +  —  • 

V^2  _^   ^2  X 

It  is  obvious  that  for  |  X  |  >  some  L2  the  right  member  is  always  negative 
since  \K'\  <  M.     If  L  is  the  larger  of  Li,  L2,  then,  since 

sm  A  =  —  1  +  — ' 

^ji2  _^  ^2  X 

and  when  |  X  |  >  Xjs; 

K^      E ^ 

^  V^2  _^  52  ' 

we  have 

0  >  sinA  >  -  1,         |X|  >  i. 

Obviously  there  are  just  two  values  of  A  in  any  [(2^•  —  l)7r,  2A-7r] 
interval  which  satisfy  this  inequality,  i.e., 

Ai  =  {2k-  i)t + iy -^  0  (^y 

A2  =  2kTr-iP+ of  y\,        0<iA<v 
where  sin  j^  = 


.'.     A  =  W7r+  (-  1)"»+V+ o/'i'j, 
for  \m\  large,  and  by  the  definition  of  A 

Xn=  m7r+  (-  l)m+i^-  f.^^^^dx-  cp+0(-), 

2  \fnj 

when   1^1   is  large;    or  since  in  each  27r-interval  there  are  2  values  of  X 
we  have 

Xn=  n7r  +  57r+  (-  l)"+«+V  -  fo'^^  dx  -  <p  +  0  (-), 

s  being  a  finite  integer. 

Case  of  C^  =  A^  -\-  B^:   Here  the  above  reasoning  does  not  apply  and 
we  attack  the  problem  by  getting  a  more  accurate  asymptotic  expansion 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 
for  D{\)  and  D'(\).     If  we  assume 


51 


v(x)  =  e^ 


'"  +  X  +  V  + 


J 


then  substituting  in  (1)  equate  coefficients,  we  are  led  to  the  expansion 


i(x)  =  Aoe^*  +  5oe-«*  +  Z) 


A^e^'+B^-^' 


t{x)  =  W  +  Doe-^'  +  Z 


A^l  X* 


or  since  we  restrict  X  to  real  values,  to 

f  \  —   A         y  I    T)    •    ^  ,   \^  Ak  cos  ^  +  Bksin  ^ 

u{x)  =  ^0  cos  ^  +  JSo  sm  ^  +  2w \t •  ' 

*=i  X* 

r  \  —  ri         >■   I    n     •     v   ,    v^  C**  COS  ^  +  D*  sin  ^ 
x{x)  =  Co  cos  ^  +  Do  sin  ^  +  2^ ^— ^ ^  > 

where  ^  is  defined  as  in  (44)  and  A,  B,  C,  D  are  independent  of  X. 
We  may  show  that  (ay)  =  (/35),  (a8)  =  (y^)  so  that 

yl  =  2(a7),         ^5  =  2(a5),         C  =  2(«/3), 

and  since  (a/9)  =  1,  we  have 

D{\) 


(105) 


V^2  _|_   ^2 


=  1  +  [2^(1)  —  «i(l)]|  cos  v?  +  Cwi(l)  +  «2(l)Ili  sin  <p, 


in  which  (wi,  vi),  (1^2,  ^2)  are  defined  by  (12)  and  <p  by  (52).     F'rom  (50) 
we  can  show  that  D'{X)  has  a  root  in  the  interval 


(2k  -  l)7r  +  iTT  <  A  <  {2k  -  l)7r  +  It. 


(106) 


By  putting  (105)  in  (1)  and  equating  coefficients  we  get  a  set  of  equations 
from  which  we  can  solve  for  A,  B,  C,  D  and  if  we  put  that  value  of  X  in 
for  which  Z)'(X)  =  0,  we  obtain 


D{\)- 


m\  +  2mQm\  cos  2ip  -\-  ml 

8X2 


wherein 


Hi) 


mi  =  \a{\)  -  \h{\),        wio  =  |a(0)  -  ^6(0). 

Hence  Z)(X)  will  have  two  roots  in  the  interval  (106)  unless 

Case  1)  Wo  =  wi  =  0; 

Case  2)     (p  =  cos~^  {ay)  =  sin~^  (a5)  =  It,     diq  =  mi; 

Case  3)    <p  =  0,    mo  =  —  mi. 


52  Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 

By  extending  the  calculation  to  the  fifth  set  of  functions  Ai,  Bi,  C4,  Di 
we  find  that  D(X)  will  still  have  two  roots  in  (106)  for  Case  1)  unless 

nil  +  2momi  cos  2^  +  m^   =  0, 
i.e.,  unless 

.1°.  m'o  =  m[  =  0; 

2°.  cp  =  Itt,     mo  =  m[; 

3°.  v'  =  0,     m,)  =  —  mi; 
in  which 

m',  ^  ia'(O)  -  i6'(0),        mi  ^  ia'(l)  -  i6'(l). 

The  same  is  true  for  Case  2)  unless 

(m;  -  m',y  +  4m^(/i  -  kY  =  0 
or 

1°.     m'l  =  mo,     mo  =  0; 
2°.     m'l  =  mo,       /i  =  lo; 
where 

/i^|a(l)  +  P(l),         /o=|a(0)+|6(0). 

It  is  probable  that  by  this  method  D{\)  would  also  be  shown  to  have 
roots  for  Case  3)  unless 

(mi  +  m,y  +  4ml{h  +  loY  =  0, 

and  that  if  one  should  continue  the  computation  one  would  find  that  D(K) 
possesses  roots  unless 

I.     mo  =  mi  =  m|  =  m,,  =  mj'  =  m^'  •  •  •  =0; 
II.       (p  =  ^TT,     mo  =  mi,     m[,  =  m[,     m[,'  =  ml',     •  •  • ; 
III.       (p  =  0,     mo  =  —  mi,     ttIq  =  —  m[,     tyiq  =  — m,",     •  •  •. 

It  is  interesting  to  note  that  for  a  restricted  case  of  I,  namely 

a{x)  ^  b{x)  ^  0, 

D(X)  possesses  only  double  roots,  provided 


(107) 


ai  =  Ti  =  1^2  =  §2  =  1, 
q;2  =  72  =  /3i  =  5i  =  0, 

Here 

2}(X)  =  2  +  2  cos  X, 

X  =  (2^  -  l)7r, 
in  fact 

^  1  +  cos  X,  sin  X 

—  sin  X,        1  +  cos  X 

This  leads  to  a  special  case  of  Fourier's  Series  in  which  the  terms  involving 
sin  mrx,  cos  nirx  for  n  even  are  wanting. 


Camp:  An  Extension  of  the  Sturm-Liouville  Expansion. 


53 


Another  interesting  result  is  the  following 
Theorem  :  A  necessary  condition  that  the  determinant 


D(\)^ 


he  of  rank  zero  is  that  (a/3)  =  (78),  where  (wi,  v\),  (ih,  V2)  are  defined  by  (12). 
We  have 

UiiuiVi)  =  ai-\-  7i?/i(l)  +  5i»i(l)  =  0, 


U2(uiVi)  =  <X2-{-  y2th{l)  +  MKl)  =  0, 
Ul{U2V2)  =  fii  +  TiW2(l)  +  8MI)  =  0, 

U2(U2V2)   =    ^2+  72«2(1)   +   52»2(1)    =   0, 

Multiply  the  first  two  equations  by  82,  —  81  respectively,  then 
(ad)  +  (t5)wi(1)  =  0. 
Similarly  using  —  a2,  «i  on  the  last  two  we  have 

(a/S)  +  (ay)u2a)  +  (a8)v2{l)  =  0. 
Again  using  72,  —  71  to  multiply  the  first  two,  we  get 

(ay)  -  (y8)vi(l)  =  0. 
Combini'ng  equations  (112),  (114),  (113)  we  obtain 


(108) 
(109) 
(110) 
(111) 

(112) 

(113) 

(114) 


Mi(l),    vx(l) 

1*2(1),       ^2(1) 


or 


(a^)  -  (yd) 

(afi)  =  (y8). 


=  0, 


Q.e.d. 


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